Chapter 12 - Numerical Methods for Constrained Optimum Design

Abstract

Basic concepts related to algorithms for constrained optimization problems are presented. Concepts of constraint status at a design point, descent function for step size calculation, and convergence requirements for an algorithm are presented. Linearization of a problem and sequential linear programming (SLP) are presented and discussed. Drawbacks of the SLP method are noted, and to overcome them, the concept of sequential quadratic programming (SQP) is introduced. This approach parallels the approach for the unconstrained optimization algorithms: calculate a search direction and a step size in that direction to determine the design change. A quadratic programming (QP) subproblem is derived for calculation of the search direction. A descent function, which is minimized to calculate a step size in the search direction, is defined. Based on this approach, an algorithm called the constrained steepest–descent (CSD) method is given for constrained optimization problems. This is a first-order method that can be viewed as an analog of the steepest–descent method but for the constrained optimization problem.